I know that the first of these is the most likely, but someone will always fail to make that assumption and interpret it differently.
Quadratic equations may be solved by factoring, but factoring when possible, finds all solutions and is not confined to quadratics.
zero multiplied by another value is zero, so the factored version shows that x=0 is also a solution.
If we divide both sides by x, we lose that particular solution, though it finds the x=2 answer.
Also remember that we really should not divide by x because we cannot divide by zero anyway,
so when we do that, we are finding only non-zero solutions.
Do check CaptainBlack's post also.